Mean square of zeta function, circle problem and divisor problem revisited

On a generalization of a theorem of Popov

“…Note that for both the Gauss circle problem and the Dirichlet divisor problem, error terms of the same quality are still wide open conjectures. We only mention that recently Bourgain and Watt [8] obtained the best known error O a 517 824 for both problems. Nevertheless, it transpires that the bound (1) is susceptible to further improvement.…”

A sharp square function estimate for the cone in $\mathbb{R}^3$

“…The paper [9] also considers a decoupling problem in which the cone is divided into small squares instead of sectors. This problem was raised by Bourgain and Watt [5] in their work on the Gauss circle problem. The paper [9] shows that the square function estimate Theorem 1.1 implies a sharp estimate for this decoupling problem.…”

A note on the Weyl formula for balls in $\mathbb{R}^d$

“…They are emerging powerful tools which have many applications especially in number theoretical problems. See for example the preprint [2] by Bourgain and Watt in which they proved (among others) improved estimates for the Dirichlet divisor and Gauss circle problems.…”